Geometric stability via information theory

TL;DR

This paper establishes stability versions of geometric inequalities like Loomis-Whitney, showing near-equality implies the shape is close to a box, using information theory, with applications to lattice isoperimetric problems.

Contribution

It introduces the first stability results for Loomis-Whitney and Uniform Cover inequalities using an information-theoretic approach, with optimal bounds up to dimension-dependent constants.

Findings

Near-equality in Loomis-Whitney implies the body is close to a box.

Stability results are optimal up to a constant depending on dimension.

Applications include stability for the edge-isoperimetric inequality in lattices.

Abstract

The Loomis-Whitney inequality, and the more general Uniform Cover inequality, bound the volume of a body in terms of a product of the volumes of lower-dimensional projections of the body. In this paper, we prove stability versions of these inequalities, showing that when they are close to being tight, the body in question is close in symmetric difference to a 'box'. Our results are best possible up to a constant factor depending upon the dimension alone. Our approach is information theoretic. We use our stability result for the Loomis-Whitney inequality to obtain a stability result for the edge-isoperimetric inequality in the infinite $d$-dimensional lattice. Namely, we prove that a subset of $\mathbb{Z}^d$ with small edge-boundary must be close in symmetric difference to a $d$-dimensional cube. Our bound is, again, best possible up to a constant factor depending upon $d$ alone.